3070
Macromolecules 1988,21, 3070-3081
(33) Llorente, M. A.; Mark, J. E. J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 181. (34) Andrady, A. L.; Llorente, M. A.; Mark, J. E. J. Chem. Phys. 1980, 73, 1439. (35) Andrady, A. L.; Llorente, M. A,; Mark, J. E. J. Chem. Phys. 1980, 72, 2282. (36) Mark, J. E.; Llorente, M. A. J. Am. Chem. SOC.1980,102,632. (37) Llorente, M. A,; Andrady, A. L.; Mark, J. E. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 621. (38) Pan, S. J.; Mark, J. E. Polym. Bull. (Berlin) 1982, 7 , 553. (39) Zhang, Z.-M.; Mark, J. E. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 473. (40) Bevington, P. B. Data and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1968; p 140. (41) Flory, P. J.; Shih, H. Macromolecules 1972,5, 761. (42) Kuwahara, N.; Okazawa, T.; Kaneko, M. J.Polym. Sci., Part C 1968,23, 543. (43) Delmas, G.; Patterson, D.; Bhattacharyya, S. N. J. Phys. Chem. 1964,68, 1468. (44) Morimoto, S. Makromol. Chem. 1970, 133, 197. (45) Speir, J. L. Adv. Organomet. Chem. 1979, 17, 407. (46) Gustavson, W. A.; Epstein, P. S.; Curtis, M. D. J. Organomet. Chem. 1982,238, 87. (47) Leung, Y. K.; Eichinger, B. E. J. Chem. Phys. 1984, 80, 3885.
(48) Ilvasky, M.; Dusek, K. Polymer 1983, 24, 981. (49) Candau, F.; Strazielle,C.; Benoit, H. Eur. Polym. J. 1976, 12, 95. (50) Flory, P. J.; Hoeve, L. A. J.; Ciferri, A. J. Polym. Sci. 1959,34, 337. (51) Flory, P. J.; Hoeve, C. A. J.; Ciferri,A. J. Polym. Sci. 1960,45, 235. (52) Flory, P. J. Macromolecules 1979, 12, 119. (53) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969. (54) Flory, P. J. J . Chem. Phys. 1977, 66, 5720. (55) Flory, P. J. Proc. R. SOC.London, A 1976, 351, 351. (56) Allen, G.; Holmes, P. A.; Walsh, D. J. Faraday Discuss. Chem. SOC.1974, 57, 19. (57) Allen, G.; Egerton, P. L.; Walsh, D. J. Polymer 1976, 17,65. (58) Flory, P. J.; Tatara, Y. J. Polym. Sci., Polym. Phys. Ed. 1975, 13, 683. (59) Flory, P. J.; Erman, B. Macromolecules 1982, 15, 800. (60) Ball, R. C.; Edwards, S. F. Macromolecules 1980, 13, 748. (61) Ball, R. C., private communication. (62) Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; Carnegie: Pittsburgh, PA, 1953; A. P. I. Project 44: (a) Table 5d; (b) Table 23a.
Integral Equation Theory of Polymer Melts: Intramolecular Structure, Local Order, and the Correlation Hole? Kenneth S. Schweizer* and John G. Curro Sandia National Laboratories, Albuquerque, New Mexico 87185. Received November 20, 1987; Revised Manuscript Received April 4, 1988
ABSTRACT: Our previously proposed microscopic, off-lattice theory of the equilibrium structure of dense polymer liquids is further developed in regard to the treatment of intramolecular polymer structure. A general scheme for self-consistently calculating the intramolecular and intermolecular pair correlations is outlined, along with the implementation of the integral equation theory for arbitrary ideal polymer models. A simple mathematical procedure for rigorously removing all unphysical intramolecular nonbonded monomer overlap is formulated in general and implemented for the freely jointed chain. The influence of the constant bond length and nonoverlapping constraints on the single-chainstructure factor is studied numerically and discussed in the context of recent small-angle neutron-scattering experiments. An extensive series of model calculations of the intermolecular radial distribution function are performed for athermal polymer melts composed of Gaussian chains, Gaussian rings, and ideal and nonoverlapping freely jointed chains. The detailed dependence of both the local, short-range order and the correlation hole on degree of polymerization, density, and intramolecular flexibility is established, along with the limiting behavior for infinite molecular weight.
I. Introduction T h e first tractable, microscopic, statistical mechanical theory for the equilibrium structure of dense one-component polymer m e l b in continuous space has been recently T h e approach employs proposed by the present a~thors.l-~ an interaction site model of polymer structure and utilizes t h e integral equation theory of Chandler and Andersen4p5 developed for small molecule fluids (the so-called “reference interaction site model” or RISM) to compute intermolecular site-site pair correlation functions. T h e high polymer problem is rendered mathematically tractable by exploiting t h e near ideality of polymers in t h e melt1v2,6-8and t h e relative unimportance of chain e n d effects for long linear macromolecules? T h e resultant theory is computationally convenient and provides a quantitative description of both long- and short-range order and density fluctuations in dense polymeric liquids. The latter features distinguish our theory from t h e incompressible random‘This work performed at Sandia National Laboratories, supported U.S.Department of Energy under Contract DE-ACO476DP00789. by the
phase approximation (RPA) approach of deGennes: which addresses only long-range correlations (small wave vector limit) of labeled species in the melt. T h e structural theory can also be employed t o calculate thermodynamic properties of the polymer fluid. In principle, our integral equation theory is applicable for arbitrary models of single polymer configurational statistics and short-range intermolecular forces. However, published applications to date1-3 have focused entirely on Gaussian chains and rings interacting via hard-core repulsions. T h e development of a reliable theory of structure a n d wave vector dependent density fluctuations in polymer melts has many applications to important physical phenomena such as X-ray scattering, neutron scattering, a n d firsborder phase transitions. In addition, the structure and thermodynamics of polymer blends appear to be sensitive to local correlations a n d compositional fluctuations as recently explicitly demonstrated via computer ~ i m u l a t i o n . ~ A realistic treatment of short-range order in the isotropic liquid requires that some degree of the chemical structure of polymer molecules must be incorporated. I n particular, a description of intramolecular structure beyond t h e 0 1988 American Chemical Society
Macromolecules, Vol. 21, No. 10, 1988
Integral Equation Theory of Polymer Melts 3071
heavily coarse-grained Gaussian model would seem to be needed. However, the relative importance of specific chemical features (e.g., bond length, bond angle, rotational isomerism, monomer shape) in determining particular observable quantities is a fascinating and largely unexplored question. For example, since low-angle neutron scattering generally probes only small wave vectors, a coarse-grained model of polymer structure may be adequate. Alternatively, wide-angle X-ray scattering is in principle sensitive to the fine details of molecular architecture. However, the simultaneous presence of multiple (chemical) length scales and conformational flexibility in polymer melts tends to wash out sharp intermolecular structural correlations even at small monomer-monomer separations and relatively low degrees of polymerization.’O This fact suggests that including all the fine details of polymeric structure may not be necessary in order to account for observable properties. As a general comment, a theoretical program that incorporates molecular detail in a stepwise fashion, and determines its consequences, would seem to be preferable from the perspective of understanding the influence of various aspects of polymeric structure on bulk liquid thermodynamic properties and correlations. The present pair of papers represents a first step toward addressing the above issues. After briefly reviewing the RISM theory of polymers in section 11, we consider the general problem of intramolecular structure in section 111. In particular, we have raised the level of chemical realism a modest amount by studying freely jointed chains in addition to their Gaussian counterparts. However, the ideality assumption still leads to unphysical nonbonded intramolecular monomer overlap, which is clearly unrealistic on very short length scales. To address this problem we develop a simple procedure for treating the intramolecular excluded volume interactions in a melt which completely removes these unphysical overlaps and leads to significant local expansion of the polymer coil. The general problem of a self-consistent treatment of intramolecular and intermolecular correlations is discussed qualitatively. An extensive series of numerical calculations for the intermolecular pair correlation function of hardcore athermal liquids composed of Gaussian chains and rings, and ideal and nonoverlapping freely jointed chains, are presented and discussed in section IV. Special attention is paid to the contact value of the radial distribution function since it plays a central role in the determination of the virial pressure.ll The detailed dependence of these properties on degree of polymerization, density, and intramolecular structure is established. The paper concludes with a brief summary of our findings. Density fluctuations, the static structure factor, and a comparison of our integral equation results with the predictions of the incompressible RPA and a continuum limit model are the subjects of the following companion paper in this issue.12
where p is the polymer molecule number density and h(r), C(r), and 4)are N X N matrices with elements hay(r), C&), and way(r),respectively. The function hay(r)= g 4 t ) - 1 is the correlated part of the intermolecular site-site radial distribution function, C,(r) is the corresponding direct correlation function between sites CY and y on different polymer molecules, and oay(r)is the normalized intramolecular probability distribution function, which for the present section is assumed known. In principle, h(r) is a nonlinear functional of o(r), and vice versa, requiring that the two sets of correlation functions be determined self-c~nsistently.~ The physical idea behind eq 1 is that the intermolecular pair correlations are propagated in a sequential fashion by “chains” of intramolecular and direct pair correlations. The direct correlation function plays a central role in modern liquid-state theory since it provides the means by which the nearly singular repulsive interactions in dense media can be theoretically handled in a nonperturbative manner. It can be interpreted as an effective or renormalized pair potential in the liquid that is a functional of both intramolecular structure and thermodynamic state.5 The set of coupled, nonlinear integral equations in eq 1 is closed by introducing a relationship (which is the approximation) between C ( r ) ,h(r), and the site-site interaction potentials. There are a variety of different schemes for accomplishing the closure depending on the type of chemical system of i n t e r e ~ t . ~ J ~For - ’ ~many nonpolar and moderately polar liquids interacting via a relatively short-range site-site potential, the packing forces dominate the s t r u ~ t u r e .In ~ this case, a hard-core interaction is a good approximation, and the corresponding closure relations are ha.,(r) = -1; < cay (24
11. RISM Integral Equation Theory The application of matrix integral RISM theory to liquids composed of high polymer chains and rings interacting via hard-core repulsions has been discussed in detail e1~ewhere.l~~ In this section we briefly summarize the relevant results. The generalized Ornstein-Zernike matrix integral equations4r5for a one-component fluid composed of polymer molecules containing N sites or chemical subunits interacting via pair-decomposable forces are h(r) = dr“ dr‘”w(lF- Fl)C(lF’- F‘l)[u(r’? + ph(r”)] (1)
This results in an enormous simplification since the matrix integral equations rigorously reduce16to a single equation given h(r) = dr“ dr“’w((F- Fl)C(lF’- F’l)[w(r”) + p,h(r’?]
1 1
J.
CJr)
r > cay
= 0;
(2b)
where cayis the distance of closest approach between sites a and y. Equation 2a is an exact statement, while eq 2b is the fundamental approximation. Equations 1 and 2 define the RISM theory of hard-core molecular f l ~ i d s . ~ ~ ~ The effects of intramolecular constraints and correlations on the intermolecular packing are explicitly taken into account by the RISM theory in an average f a ~ h i o n . ’ ~ ~ ~ ~ For high polymers a direct numerical solution of the N ( N + 1)/4 coupled, nonlinear integral equations is intractable. We therefore have considered polymer rings and developed an approximation scheme for linear chains composed of identical monomers (sites). A. Ring Polymers. As an obvious consequence of the topological symmetry of a ring, all intermolecular correlation functions in eq 1 are equivalent on average ha+) = h(r) = g ( r ) - 1
C&)
= C(r)
(3)
1 1
(4) where p m = Np is the monomer or site density and w ( r ) is the single-ring polymer structure factor a=l
The closure relations for eq 4 are the site-index inde-
Macromolecules, Vol. 21, No. 10, 1988
3072 Schweizer and Curro
pendent versions of eq 2. Generalization of the integral equation theory to polymers composed of a small number of chemically distinct sites or monomers is straightforward. B. Linear Chains. A computationally tractable approach to the linear chain problem has been recently developed3 by us based on the simple idea that chain end effects are expected to be small for large N. The detailed development is discussed elsewhere: but the idea is to take the linear chain structure into account in an average fashion neglecting the explicit chain end effects, i.e., h,(r) = h(r)and Ca7(r)= C(r) for all a and y. In the context of our integral equation theory such an approximation can be formulated in an optimum manner, which minimizes errors in collective quantities such as &h&). The result is a single integral equation identical in form with the ring polymer case (eq 4) but where w ( r ) is the singlechain structure factor
situations for which the ideality theorem either incurs quantitative errors or may break down entirely. We will discuss these subsequently, but first we summarize various “ideal” models that can be studied with our integral equation theory and their corresponding intramolecular structure factors. For Gaussian ring polymers, with statistical segment length d, the intramolecular structure factor is192,z3 N-1
O(k) = 1
+ 2N-1 t=1 C ( N - t) exp[-k2d2t(N - t)/6N]
and the sum must be computed numerically. For Gaussian and freely jointed chains the structure factor can be analyticallyz4calculated with the result O ( k ) = (1 - f)-2[1 - f. - 2N4f + 2N-’fN+’] (9) where
N
w ( r ) 6 N-l
C
a&)
f = exp(-k2dz/6)
(6)
a,y=l
Systematic and mathematically tractable procedures to calculate corrections to the chain-averaged theory have been pre~ented.~ However, for the collective or averaged radial distribution function defined as
(8)
for the Gaussian chain and f = sin ( k l o ) / k l o
(10) (11)
for a freely jointed chain with a fixed bond length lo. Semiflexible wormlike chains can also be treated approxi m a t e l ~ .Finally, ~~ at the expense of significantly more N numerical effort, the intramolecular structure factor of g(r) E N-2 C g&) (7) rotational isomeric state chains can in principle be calcun,y=l lated by either Monte Carlo simulation and/or moment a large degree of cancellation of errors is expected. This expansions. follows from the fact that the leading order correction to The use of ideal forms for G ( k ) corresponds to the asour approximation scheme3 for g(r) vanishes for all N. sumption that intramolecular excluded volume effects can Many properties of experimental interest, such as the be ignored on all length scales and for all liquid densities. equation-of-state in the high polymer limit1’ and the static Mathematically, this affords a great simplification in the structure factor of neat melts, depend only on collective implementation of the RISM theory since O ( k ) can be or “doubly-summed” correlation functions. Equations 2, calculated separately without having to determine it 4, and 5 (or 6) can be solved numerically by utilizing self-consistently with the intermolecular correlations. standard procedures discussed e l s e ~ h e r e . ~ ~ ~ ~ ~ * ’ ~ Indeed, a direct treatment of the intramolecular excluded volume problem for high polymers which is valid for all 111. Intramolecular Structure (not just large) length scales of finite size monomers (not There are two distinct aspects related to the treatment &functionpseudopotentiakP) does not exist. Nevertheless, of the intramolecular structure of polymers in the melt. it is clear that intramolecular excluded volume can never The first pertains to the level of chemical detail and conbe ignored at very small separations since nonbonded stitutes the adoption of a mathematical model that specmonomers on the same chain cannot overlap, and this will ifies the short-range interactions between nearby monoalways have local consequences even if it is negligible on mers and associated bonding constraints. Typically, the large length scales. This consideration represents the first model can vary widely in complexity all the way from the problem with the literal use of the ideal description. The highly simplified Gaussian model up to a chemically resecond fundamental problem with the ideal description is alistic rotational isomeric state description.18 The second its neglect of a self-consistent link between the intramofundamental aspect involves the treatment of the nonlecular and intermolecular correlation^.^ Such coupling bonded intramolecular excluded volume interactions and is inherently nonlinear and can produce a variety of their self-consistent modification by the presence of other “nonideal” phenomena such as condensed-phase modifipolymer molecules in the dense melt. Both these issues cation of isomeric e q ~ i l i b r i a ,locally ~ * ~ ~parallel chain domust be addressed in order to implement the RISM inmain structure in high-density /low-temperature liquids tegral equation theory. composed of stiff polymers,31and collapse or expansion A. Ideal Limit. In our previous ~ o r k l the - ~ simplest of chain dimensions with respect to the unperturbed possible approaches to the above two issues were adopted: The formulation of a microscopic statistical ideal Gaussian chains and rings were studied. The ideality mechanical theory to address these issues is qualitatively principle of Flory’ states that for flexible polymers in the sketched in the next subsection. melt the intramolecular configuration is determined enB. Self-consistent Treatment of Intramolecular tirely by short-range effects; i.e., the strong intra- and and Intermolecular Correlations. 1. General Forintermolecular excluded volume interactions effectively mulation. A general classical statistical mechanical theory cancel. A litany of both neutron-scattering experiments of the structure of flexible molecules in condensed phases on labeled chains8 and computer s i m u l a t i ~ n s ~have ~J~ has been developed by Chandler and Pratt.3z In our nodocumented the fundamental correctness of Flory’s ideas, tation, their fundamental result is an expression for the including the noteworthy fact that the configurations of full N-body intramolecular distribution function, w[{R)], monomer sequences much shorter (as small as 10 A) than of a tagged molecule immersed in a liquid the global chain dimension obey ideal statistics unper4 R l I %[(RIlY[(RJl (12) turbed by intramolecular or intermolecular excluded where (R)denotes a complete set of coordinates necessary volume interactiomWz2There are, of course, systems and 18926,27
Macromolecules, Vol. 21, No. 10, 1988 to specify a particular polymer configuration, wo[(R]]is the unperturbed (single molecule) distribution function, and y[(R)] is the cavity distribution function or influence functional for a polymer in the The latter quantity describes the condensed-phase modification of the intramolecular potential surface and is the Boltzmann factor for the reversible work associated with altering configurations of the polymers, i.e. In Y W 1 1 = -0AP[(R11 (13) where Ap[(R)] is the solvent (i.e., surrounding polymer molecules) induced part of the potential of mean force. The unperturbed distribution function can also be expressed (within classical mechanics) as a Boltzmann factor of a “bare” potential energy which can be separated into its short-range, “ideal” part, Uo,and a part describing the long-range intramolecular excluded volume interactions, UE, which is generally taken to be pair decomposable. Therefore, the full N-body distribution function can be written as an exponential of a potential of a mean force: w(R) a exp(-PW) where (14) W[(R)] E Uo + U, + Ap If one has an explicit expression for the condensed-phase component, Ap, then in principle the effective intramolecular configurational statistics problem defined by eq 14 can be solved, and the corresponding pair or two-body functions (way(r)irequired by the RISM theory can be obtained. However, in practice Ap is exceedingly complex and not even pair dec~mposable,~~ reflecting the fact that the potential of mean force associated with one pair of sites depends on the configuration of many other sites. This physical feature has been discussed in detail by Chandler et a1.33134within the context of the quantum solvated electron problem and the path-integral formulation of quantum mechanics.35 As emphasized by Chandler, their analysis is quite general and is applicable to real classical, flexible polymeric systems. A self-consistent mean field scheme has been developed to address this problem.% The basic result is that the solvent-induced interactions are approximated as pair decomposable but must be determined self-consistently since they are a functional of both the intramolecular and intermolecular pair correlations functions themselves, (way(r))and (Cay(r)).In the resulting effective intramolecular statistics problem, U, and Ap are then both pair decomposable, but a very formidable the\ oretical problem remains since one has to simultaneously grapple with the short-range (ideal) interactions, intramolecular excluded volume, and a complex solvent-induced potential which itself must be determined self-consistently. A tractable approximate solution to this problem for real polymers can be formulated based on optimized perturbation theory about a suitably chosen reference but this is beyond the scope of the present paper. In this initial work, we shall adopt a much simpler approach guided by simple screening ideas. 2. Limiting Approximation. For neat melts the fundamental feature of the pair-decomposable, self-consistently determined solvent-induced potential is that it is strongly attractive at small separations between pairs of polymer sites, thereby favoring a collapsed polymer structure.u This qualitative behavior is a consequence of the physical fact that by shrinking the tagged (solute) polymer less solvent free energy change is needed to incorporate it in the dense liquid. This trend is enhanced with increasing solvent density and implies that the intramolecular excluded volume and solvent-induced potential tend to cancel in eq 14, at least in an average or integrated sense.
Integral Equation Theory of Polymer Melts 3073
If the “solvent”-inducedscreening of the intramolecular excluded volume interactions is almost complete for separations on the order of a monomer size (as is expected for a dense liquid), then a simple zero-order scheme for the treatment of intramolecular excluded volume of polymers in the melt can be envisioned: the intramolecular polymer structure can be treated as ideal except for separations less than or equal to the monomer diameter. In other words, for hard-core interactions, nonbonded monomer overlap is rigorously forbidden, but the intramolecular excluded volume has no other consequences. Such an approach would appear to be the simplest nontrivial treatment of nonideality. C. Elimination of Intramolecular Overlap. The simple approximation discussed above corresponds mathematically to adopting the ideal values for the intramolecular distribution functions (o,(r)] for separations of nonbonded pairs beyond the contact value and requiring vanishing probability for overlapping configurations. Such a scheme represents a non-Hamiltonian-based approximation and avoids entirely the self-consistency issues and a possible density dependence of the intramolecular structure. Clearly, the molecular weight scaling of the global dimensions of the polymer will still obey ideal random walk statistics, although both long-range and local expansion of the polymer coil over its purely ideal dimensions will occur. In the remainder of this paper we will concentrate on the freely jointed chain, and hence the relevant equations associated with the implementation of the above scheme will be presented only for this case. Analogous results for other ideal polymer statistics models are straightforwardly obtained in principle. The Fourier transform of the intramolecular probability distribution function between sites a and y of the ideal freely jointed chain is24 Oa,(k) = [sin (kl)/kl]la-71
(15)
where 1 is the fixed bond length. For simplicity, in all our numerical calculations on hard-core chains we chose 1 = a = hard-core diameter. The treatment of more general (1 f a ) “bead-rodn models is straightforward. Our nonoverlapping freely jointed chain model is defined by
way(r) = 0; r
